Achiral hard bananas assemble double-twist skyrmions and blue phases

The study investigates whether achiral, hard banana-shaped particles with purely excluded-volume interactions can spontaneously assemble skyrmions and blue phases via bend-flexoelectric coupling. Skyrmions, historically associated with chiral interactions in magnets and chiral liquid crystals, are topologically protected vortex-like configurations. In liquid crystals, highly chiral systems form cholesteric phases and blue phases comprising double-twist cylinders (skyrmion filaments). Separately, achiral bent-core mesogens are known to form twist-bend nematic phases through bend-flexoelectric effects. Motivated by these parallels, the central question is whether bend-induced coupling in achiral curved particles can stabilize skyrmion-like structures and blue phases, under confinement and in bulk, and how their size and dynamics depend on particle geometry and external fields. The purpose is to reveal new routes to skyrmion stabilization without intrinsic chirality, with potential relevance for reconfigurable materials, photonics, and memory devices.

Background covers the historical development of skyrmions from nuclear physics to condensed matter and their typical emergence from chiral interactions in helical ferromagnets and chiral nematic (cholesteric) liquid crystals. In cholesterics, blue phases (BPI–III) arise from the local preference for double-twist cylinders (DTCs) that cannot tile space without defects. BPI and BPII are cubic DTC lattices; BPIII is an amorphous, thermally fluctuating tangle of DTCs ("blue fog"). Confinement can align DTC axes normal to walls, yielding quasi-2D baby skyrmions and skyrmion lattices. Independently, achiral bent-core systems exhibit twist-bend (NTB) and splay-bend (NSB) nematics due to bend-flexoelectric coupling, with recent simulations and experiments confirming such phases for colloidal bananas and bent rods. Theoretical works proposed that bend-flexoelectricity could stabilize complex modulated phases, including polar analogs of blue phases, but evidence in particle-based simulations and experiments for bend-stabilized skyrmions has been limited; a prior 2D vortex phase in polydisperse bananas did not constitute true skyrmions. This work addresses that gap by demonstrating skyrmion and BPIII formation from achiral bananas.

Particle model and interactions: Each banana-shaped particle is a rigid arc of 17 tangent spherical beads (diameter σ) with contour length L = 16σ and radius of curvature R, defining opening angle ψ = L/R and length-to-diameter ratio L/σ = 16. Beads interact via a pseudo hard-sphere, cut-and-shifted Mie potential tuned to reproduce hard-sphere thermodynamics and structure at reduced temperature kB T/ε = 1.5 (parameters λ1=50, λ2=49; remaining cut/shift as in cited model). Units: length σ, energy ε, time τMD = σ√(m/ε). Dynamics: Velocity-Verlet integration with timestep Δt = 0.0035 τMD. Thermostat/barostat: Langevin thermostat (damping 0.005 τMD−1) and Berendsen barostat (coupling 200 τMD) for NPT runs. Software: LAMMPS. Bulk simulations: Systems with N = 1352 up to 9000 bananas in orthorhombic boxes with 3D periodic boundaries. NPT ensemble to map equation of state and global order parameters (nematic S and heliconical T). The I → NTB transition region was scanned via compression/expansion protocols to probe for intermediate structures. Confinement simulations: Two parallel, structureless walls with surface normals along z modeled by truncated-shifted harmonic potentials of strength cw = 0.1 kB T/σ2 at z1 and z2 (Ht = z2 − z1). Initial isotropic monolayer (N ≈ 4500) laterally compressed at βPσ3 = 0.05 to target effective packing fraction η ≈ 0.33, then equilibrated in NVT up to 1010 MD steps. Effective confinement height H determined from bead density profiles. Additional periodic-confinement setup imposed H/Lee ≈ 1.5 (Lee = 2R sin(ψ/2)) to stabilize a hexagonal skyrmion lattice across periodic boundaries. Structure identification and order parameters: Local nematic order S and heliconical order T computed from local environments using Q and M2 tensors; thresholds S < 0.18 and |T| > 0.3 identify bananas belonging to DTCs (skyrmion filaments) and provide handedness via sign(T). Skyrmion size rs measured from the banana–skyrmion orientational correlation S220(r) = ⟨3(ui·uv)2 − 1⟩ normalized by S000, where r is orthogonal distance to the filament axis; rs taken at the first minimum of S220(r). Orientation profiles map average tilt angle θ(r) from S220 values. Dynamics: Computed single-particle orientational correlation Cα(t) = ⟨u(0)·u(t)⟩ and relaxation time τ0, mean square displacement MSD(t) with components parallel (MSD∥) and perpendicular (MSD⊥) to the initial particle axis, self-intermediate scattering function Fs(q,t) at various q, and non-Gaussian parameter α2(t) to assess dynamic heterogeneity. Field control: Prototype MD with N ≈ 1000 bananas including induced dipole along particle long axis subject to a square-wave electric field (period ≈ 1.5/τMD, amplitude β μ0 E = 3.3) applied parallel to skyrmion axes to switch between skyrmion and uniaxial nematic states.

• Achiral, hard banana-shaped particles spontaneously form bend-stabilized skyrmions and blue phases driven by entropy and bend-flexoelectric coupling, without intrinsic chiral interactions.
• Confinement phase sequence (ψ = 1.6, L/σ = 16, η ≈ 0.33) versus height H: splay-bend-like (SB) for H < Lee; isolated skyrmions (IS) for Lee < H < 2 rs; hexagonal skyrmion lattice (SL) at larger H; and a confined BPIII-like phase near the equilibrium pitch.
• Skyrmion filaments under confinement are racemic (mixtures of left- and right-handed), forming a quasi-2D hexagonal lattice because the inter-skyrmion interaction is achiral for half-skyrmions; slight handedness imbalance arises from finite-size fluctuations.
• Structural characterization confirms half-skyrmions (merons): the director rotates by π/2 from core to boundary; polarization field P(r) aligns with bend vector and is undefined on the filament axis (β-line). Orientation correlations yield S220(rs) ≈ −0.68 at the first minimum (tilt ≈ 85° at boundary) and S220(0) ≈ 1.28 (tilt ≈ 35° at center), independent of ψ within the studied range.
• Tunable skyrmion size: rs decreases with increasing opening angle ψ and scales with particle curvature R(ψ) = L/ψ, i.e., rs(ψ) ~ R(ψ). Data for SL at ψ = 1.2, 1.3, 1.6 and for BPIII at ψ = 1.6 collapse on this trend.
• Dynamics in SL and BPIII are fluid-like with non-Gaussian features: orientational relaxation τ0 ≈ 10^1 τMD; MSD shows diffusive long-time behavior with MSD⊥ surpassing MSD∥ at long times and a cage-like plateau in MSD⊥ at intermediate times; Fs(q,t) exhibits a single-step structural relaxation (no pronounced two-step decay); α2(t) displays a peak at intermediate times, indicating dynamic heterogeneity.
• Electric-field control: A square-wave field (period ≈ 1.5/τMD, β μ0 E = 3.3) switches the system reversibly between a skyrmion phase (field off) and a uniaxial nematic aligned along the field (field on), demonstrating robustness and reconfigurability.
• Bulk behavior near the I–NTB transition: In small systems (N = 1352), transient double-twist vortices appear during I↔NTB transformation but vanish at equilibrium. In larger systems (N = 9000), a disordered network of flexible DTCs nucleates from the isotropic phase within ~10^7 τMD and remains stable for at least 2×10^8 τMD, consistent with a BPIII composed of half-skyrmion filaments embedded in an isotropic background.
• BPI and BPII cubic lattices were not observed within the explored parameter space, consistent with short NTB pitch and strong fluctuations destabilizing crystalline order; aligns with prior theoretical and experimental reports in low-pitch chiral systems.

The findings directly address the central hypothesis that bend-flexoelectric coupling in achiral curved particles can stabilize skyrmionic textures and blue phases. Under confinement, frustration of the NTB pitch drives the formation of rigid, racemic half-skyrmions, assembling into isolated vortices or hexagonal lattices depending on film thickness. In bulk near the I–NTB transition, local preference for bend and double-twist yields an amorphous network of flexible skyrmion filaments identified as BPIII. The consistent half-skyrmion twist angle and the scaling of skyrmion radius with particle curvature demonstrate geometry-controlled skyrmion size, decoupled from intrinsic molecular chirality. Dynamically, both SL and BPIII behave as non-Gaussian fluids, reconciling fast structural relaxation with heterogeneous single-particle motion. The ability to reversibly create and annihilate skyrmions via external fields showcases their robustness and responsiveness, highlighting potential for tunable electro-optic and information-storage applications. Overall, the results extend the paradigm of skyrmion stabilization beyond chiral interactions, revealing entropy- and shape-driven routes to complex topological structures in soft matter.

This work demonstrates, via extensive MD simulations, that achiral hard banana-shaped particles stabilize half-skyrmion filaments and a BPIII-like phase through bend-flexoelectric coupling. Under thin confinement, the system forms racemic isolated skyrmions or hexagonal skyrmion lattices; in bulk near the I–NTB transition, it assembles a dynamic network characteristic of BPIII. Skyrmion size is tunable and scales with particle curvature, and the structures exhibit fluid-like yet non-Gaussian dynamics. External electric fields can reversibly switch between skyrmion and nematic states. These results open routes to designing reconfigurable topological soft materials without intrinsic chirality, with implications for photonics, displays, and memory devices. Future work should broaden parameter scans (particle geometry, density, polydispersity), explore alternative boundary/anchoring conditions, quantify energy landscapes and defect kinetics, investigate other external-field protocols, and seek experimental realizations in thermotropic and lyotropic systems.

• Stability window: The BPIII appears thermodynamically stable only within an extremely narrow density range near the I–NTB transition.
• Finite size and time: Formation and equilibration of modulated bulk phases require large N and long runs; smaller systems suppressed stable skyrmions due to finite-size and periodic self-interaction effects. Despite long simulations, complete exploration of phase space remains limited.
• Parameter space: The study focuses on a narrow set of particle parameters (L/σ = 16, ψ primarily 1.6, with some 1.2–1.3 in SL) and interaction models; other geometries or interactions might support different blue phases.
• Absence of BPI/BPII: While not observed here, the simulations cannot definitively exclude their emergence under other conditions (e.g., longer pitch, altered anchoring, different densities or temperatures).
• Minor artifacts: Slight handedness imbalance in SL attributed to finite-size fluctuations; some reported dynamic times are in reduced units, complicating direct experimental timescale comparison.